# Fresnel integral

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The Fresnel integralsS(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:

A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.

Augustin-Jean Fresnel was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular theory, from the late 1830s  until the end of the 19th century. He is perhaps better known for inventing the catadioptric (reflective/refractive) Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea. The simpler dioptric stepped lens, first proposed by Count Buffon  and independently reinvented by Fresnel, is used in screen magnifiers and in condenser lenses for overhead projectors.

Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

## Contents

${\displaystyle S(x)=\int _{0}^{x}\sin(t^{2})\,dt,\quad C(x)=\int _{0}^{x}\cos(t^{2})\,dt.}$

The simultaneous parametric plot of S(x) and C(x) is the Euler spiral (also known as the Cornu spiral or clothoid). Recently, they have been used in the design of highways and other engineering projects. [1]

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object.

An Euler spiral is a curve whose curvature changes linearly with its curve length. Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals.

## Definition

The Fresnel integrals admit the following power series expansions that converge for all x:

{\displaystyle {\begin{aligned}S(x)&=\int _{0}^{x}\sin(t^{2})\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+3}}{(2n+1)!(4n+3)}}.\\C(x)&=\int _{0}^{x}\cos(t^{2})\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+1}}{(2n)!(4n+1)}}.\end{aligned}}}

Some authors, including Abramowitz and Stegun, (eqs 7.3.1 7.3.2) use ${\displaystyle \pi t^{2}/2}$ for the argument of the integrals defining S(x) and C(x). This changes their limits at infinity from ${\displaystyle \textstyle (1/2)\cdot {\sqrt {\pi /2}}}$ to ${\displaystyle 1/2}$ and the arc length for the first spiral turn ${\displaystyle (2\pi )}$ to 2 (at ${\displaystyle t=2}$), all smaller by a factor ${\displaystyle \textstyle {\sqrt {2/\pi }}}$. The alternative functions are also called Normalized Fresnel integrals.

Abramowitz and Stegun (AS) is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the National Institute of Standards and Technology (NIST). Its full title is Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. A digital successor to the Handbook was released as the "Digital Library of Mathematical Functions" (DLMF) on May 11, 2010, along with a printed version, the NIST Handbook of Mathematical Functions, published by Cambridge University Press.

## Euler spiral

The Euler spiral , also known as Cornu spiral or clothoid, is the curve generated by a parametric plot of ${\displaystyle S(t)}$ against ${\displaystyle C(t)}$. The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering.

In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.

Marie Alfred Cornu was a French physicist. The French generally refer to him as Alfred Cornu.

A nomogram, also called a nomograph, alignment chart or abaque, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. The field of nomography was invented in 1884 by the French engineer Philbert Maurice d’Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision. Nomograms use a parallel coordinate system invented by d'Ocagne rather than standard Cartesian coordinates.

From the definitions of Fresnel integrals, the infinitesimals ${\displaystyle dx}$ and ${\displaystyle dy}$ are thus:

In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.

{\displaystyle {\begin{aligned}dx&=C'(t)\,dt=\cos(t^{2})\,dt,\\dy&=S'(t)\,dt=\sin(t^{2})\,dt.\end{aligned}}}

Thus the length of the spiral measured from the origin can be expressed as

${\displaystyle L=\int _{0}^{t_{0}}{\sqrt {dx^{2}+dy^{2}}}=\int _{0}^{t_{0}}dt=t_{0}.}$

That is, the parameter ${\displaystyle t}$ is the curve length measured from the origin ${\displaystyle (0,0)}$, and the Euler spiral has infinite length. The vector ${\displaystyle (\cos(t^{2}),\sin(t^{2}))}$ also expresses the unit tangent vector along the spiral, giving ${\displaystyle \theta =t^{2}}$. Since t is the curve length, the curvature ${\displaystyle \kappa }$ can be expressed as

${\displaystyle \kappa ={\frac {1}{R}}={\frac {d\theta }{dt}}=2t.}$

And the rate of change of curvature with respect to the curve length is

${\displaystyle {\frac {d\kappa }{dt}}={\frac {d^{2}\theta }{dt^{2}}}=2.}$

An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering: If a vehicle follows the spiral at unit speed, the parameter ${\displaystyle t}$ in the above derivatives also represents the time. That is, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.

Sections from Euler spirals are commonly incorporated into the shape of roller-coaster loops to make what are known as clothoid loops.

## Properties

• C(x) and S(x) are odd functions of x.
• Asymptotics of the Fresnel integrals as ${\displaystyle x\to \infty }$ are given by the formulas:
{\displaystyle {\begin{aligned}S(x)&={\sqrt {\frac {\pi }{2}}}\left({\frac {{\mbox{sign}}{(x)}}{2}}-\left[1+O(x^{-4})\right]\left({\frac {\cos {(x^{2})}}{x{\sqrt {2\pi }}}}+{\frac {\sin {(x^{2})}}{x^{3}{\sqrt {8\pi }}}}\right)\right),\\C(x)&={\sqrt {\frac {\pi }{2}}}\left({\frac {{\mbox{sign}}{(x)}}{2}}+\left[1+O(x^{-4})\right]\left({\frac {\sin {(x^{2})}}{x{\sqrt {2\pi }}}}-{\frac {\cos {(x^{2})}}{x^{3}{\sqrt {8\pi }}}}\right)\right).\end{aligned}}}
• Using the power series expansions above, the Fresnel integrals can be extended to the domain of complex numbers, and they become analytic functions of a complex variable.
• The Fresnel integrals can be expressed using the error function as follows: [2]
{\displaystyle {\begin{aligned}S(z)&={\sqrt {\frac {\pi }{2}}}{\frac {1+i}{4}}\left[\operatorname {erf} \left({\frac {1+i}{\sqrt {2}}}z\right)-i\operatorname {erf} \left({\frac {1-i}{\sqrt {2}}}z\right)\right],\\C(z)&={\sqrt {\frac {\pi }{2}}}{\frac {1-i}{4}}\left[\operatorname {erf} \left({\frac {1+i}{\sqrt {2}}}z\right)+i\operatorname {erf} \left({\frac {1-i}{\sqrt {2}}}z\right)\right].\end{aligned}}}
or
{\displaystyle {\begin{aligned}C(z)+iS(z)&={\sqrt {\frac {\pi }{2}}}{\frac {1+i}{2}}\operatorname {erf} \left({\frac {1-i}{\sqrt {2}}}z\right),\\S(z)+iC(z)&={\sqrt {\frac {\pi }{2}}}{\frac {1+i}{2}}\operatorname {erf} \left({\frac {1+i}{\sqrt {2}}}z\right).\end{aligned}}}

### Limits as x approaches infinity

The integrals defining C(x) and S(x) cannot be evaluated in the closed form in terms of elementary functions, except in special cases. The limits of these functions as x goes to infinity are known:

${\displaystyle \int _{0}^{\infty }\cos t^{2}\,dt=\int _{0}^{\infty }\sin t^{2}\,dt={\frac {\sqrt {2\pi }}{4}}={\sqrt {\frac {\pi }{8}}}.}$

The limits of C and S as the argument tends to infinity can be found by the methods of complex analysis. This uses the contour integral of the function

${\displaystyle e^{-t^{2}}}$

around the boundary of the sector-shaped region in the complex plane formed by the positive x-axis, the bisector of the first quadrant y = x with x ≥ 0, and a circular arc of radius R centered at the origin.

As R goes to infinity, the integral along the circular arc ${\displaystyle \gamma _{2}}$ tends to 0

${\displaystyle \left|\int _{\gamma _{2}}e^{-t^{2}}dt\right|\leq \int _{\gamma _{2}}|e^{-t^{2}}|dt=R\int _{0}^{\pi /4}e^{-R^{2}\cos 2t}dt\leq R\int _{0}^{\pi /4}e^{-R^{2}(1-{\frac {4}{\pi }}t)}dt={\frac {\pi }{4R}}\left(1-e^{-R^{2}}\right),}$

where polar coordinates ${\displaystyle z=Re^{it}}$ were used and Jordan's inequality was utilised for the second inequality. The integral along the real axis ${\displaystyle \gamma _{1}}$ tends to the half Gaussian integral

${\displaystyle \int _{0}^{\infty }e^{-t^{2}}\,dt={\frac {\sqrt {\pi }}{2}}.}$

Note too that because the integrand is an entire function on the complex plane, its integral along the whole contour is zero. Overall, we must have

${\displaystyle \int _{0}^{\infty }e^{-t^{2}}\,dt=\int _{\gamma _{3}}e^{-t^{2}}\,dt,}$

where ${\displaystyle \gamma _{3}}$ denotes the bisector of the first quadrant, as in the diagram. To evaluate the right hand side, parametrize the bisector as

${\displaystyle t=re^{\pi i/4}={\frac {\sqrt {2}}{2}}(1+i)r}$

where r ranges from 0 to ${\displaystyle +\infty }$. Note that the square of this expression is just ${\displaystyle +ir^{2}}$. Therefore, substitution gives the right hand side as

${\displaystyle \int _{0}^{\infty }e^{-ir^{2}}{\frac {\sqrt {2}}{2}}(1+i)\,dr.}$

Using Euler's formula to take real and imaginary parts of ${\displaystyle e^{-ir^{2}}}$ gives this as

{\displaystyle {\begin{aligned}&\int _{0}^{\infty }(\cos(r^{2})-i\sin(r^{2})){\frac {\sqrt {2}}{2}}(1+i)\,dr\\={}&{\frac {\sqrt {2}}{2}}\int _{0}^{\infty }\left[\cos(r^{2})+\sin(r^{2})+i\left(\cos(r^{2})-\sin(r^{2})\right)\right]\,dr={\frac {\sqrt {\pi }}{2}}+0i,\end{aligned}}}

where we have written ${\displaystyle 0i}$ to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting ${\displaystyle I_{C}=\int _{0}^{\infty }\cos(r^{2})\,dr,I_{S}=\int _{0}^{\infty }\sin(r^{2})\,dr}$ and then equating real and imaginary parts produces the following system of two equations in the two unknowns ${\displaystyle I_{C},I_{S}}$:

{\displaystyle {\begin{aligned}I_{C}+I_{S}&={\sqrt {\frac {\pi }{2}}},\\I_{C}-I_{S}&=0.\end{aligned}}}

Solving this for ${\displaystyle I_{C}}$ and ${\displaystyle I_{S}}$ gives the desired result.

## Generalization

The integral

${\displaystyle \int x^{m}\exp(ix^{n})\,dx=\int \sum _{l=0}^{\infty }{\frac {i^{l}x^{m+nl}}{l!}}\,dx=\sum _{l=0}^{\infty }{\frac {i^{l}}{(m+nl+1)}}{\frac {x^{m+nl+1}}{l!}}}$
{\displaystyle {\begin{aligned}\int x^{m}\exp(ix^{n})\,dx&={\frac {x^{m+1}}{m+1}}\,_{1}F_{1}\left({\begin{array}{c}{\frac {m+1}{n}}\\1+{\frac {m+1}{n}}\end{array}}\mid ix^{n}\right)\\&={\frac {1}{n}}i^{(m+1)/n}\gamma \left({\frac {m+1}{n}},-ix^{n}\right),\end{aligned}}}

which reduces to Fresnel integrals if real or imaginary parts are taken:

${\displaystyle \int x^{m}\sin(x^{n})\,dx={\frac {x^{m+n+1}}{m+n+1}}\,_{1}F_{2}\left({\begin{array}{c}{\frac {1}{2}}+{\frac {m+1}{2n}}\\{\frac {3}{2}}+{\frac {m+1}{2n}},{\frac {3}{2}}\end{array}}\mid -{\frac {x^{2n}}{4}}\right)}$.

The leading term in the asymptotic expansion is

${\displaystyle _{1}F_{1}\left({\begin{array}{c}{\frac {m+1}{n}}\\1+{\frac {m+1}{n}}\end{array}}\mid ix^{n}\right)\sim {\frac {m+1}{n}}\,\Gamma \left({\frac {m+1}{n}}\right)e^{i\pi (m+1)/(2n)}x^{-m-1},}$

and therefore

${\displaystyle \int _{0}^{\infty }x^{m}\exp(ix^{n})\,dx={\frac {1}{n}}\,\Gamma \left({\frac {m+1}{n}}\right)e^{i\pi (m+1)/(2n)}.}$

For m = 0, the imaginary part of this equation in particular is

${\displaystyle \int _{0}^{\infty }\sin(x^{a})\,dx=\Gamma \left(1+{\frac {1}{a}}\right)\sin \left({\frac {\pi }{2a}}\right),}$

with the left-hand side converging for a > 1 and the right-hand side being its analytical extension to the whole plane less where lie the poles of ${\displaystyle \Gamma (a^{-1})}$.

The Kummer transformation of the confluent hypergeometric function is

${\displaystyle \int x^{m}\exp(ix^{n})\,dx=V_{n,m}(x)e^{ix^{n}},}$

with

${\displaystyle V_{n,m}:={\frac {x^{m+1}}{m+1}}\,_{1}F_{1}\left({\begin{array}{c}1\\1+{\frac {m+1}{n}}\end{array}}\mid -ix^{n}\right).}$

## Numerical approximation

For computation to arbitrary precision, the power series is suitable for small argument. For large argument, asymptotic expansions [4] converge faster. Continued fraction methods may also be used. [5]

For computation to particular target precision, other approximations have been developed. Cody [6] developed a set of efficient approximations based on rational functions that give relative errors down to 2×10−19. A FORTRAN implementation of the Cody approximation that includes the values of the coefficients needed for implementation in other languages was published by van Snyder. [7] Boersma developed an approximation with error less than 1.6×10−9. [8]

## Applications

The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects. [9] More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see track transition curve. [1] Other applications are roller coasters [9] or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.[ citation needed ]

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## References

1. James Stewart (2008). Calculus Early Transcendentals. Cengage Learning EMEA. p. 383. ISBN   978-0-495-38273-7.
2. functions.wolfram.com, Fresnel integral S: Representations through equivalent functions and Fresnel integral C: Representations through equivalent functions. Note: Wolfram uses the Abramowitz & Stegun convention, which differs from the one in this article by factors of ${\displaystyle {\sqrt {\pi /2}}}$
3. Mathar, R. J. (2012). "Series Expansion of Generalized Fresnel Integrals". arXiv: [math.CA].
4. Temme, N. M. (2010), "Error Functions, Dawson's and Fresnel Integrals: Asymptotic expansions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN   978-0521192255, MR   2723248
5. Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 6.8.1. Fresnel Integrals". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN   978-0-521-88068-8.
6. Cody, William J (1968). "Chebyshev approximations for the Fresnel integrals" (PDF). Math. Comp. 22 (102): 450–453. doi:10.1090/S0025-5718-68-99871-2.
7. van Snyder, W (December 1993). "Algorithm 723: Fresnel integrals". ACM Trans. Math. Softw. 19 (4): 452–456. doi:10.1145/168173.168193.
8. Boersma, J. (1960). "Computation of Fresnel Integrals". Math. Comp. 14 (72): 380. doi:10.1090/S0025-5718-1960-0121973-3. MR   0121973.
9. Beatty, Thomas. "How to evaluate Fresnel Integrals" (PDF). FGCU MATH - SUMMER 2013. Retrieved 27 July 2013.